Gluing and Hilbert functions of monomial curves

Abstract

In this article, by using the technique of gluing semigroups, we give infinitely many families of 1-dimensional local rings with non-decreasing Hilbert functions. More significantly, these are local rings whose associated graded rings are not necessarily Cohen-Macaulay. In this sense, we give an effective technique for constructing large families of 1-dimensional Gorenstein local rings associated to monomial curves, which support Rossi's conjecture saying that every Gorenstein local ring has a non-decreasing Hilbert function.

Cohen-Macaulay Criteria for Projective Monomial Curves via Gröbner Bases

“…Next we prove that (a),(b) ⇒ (c). From the proof above of the equivalence (a) ⇐⇒ (b), we see that under the assumption that (a) (hence also (b)) holds, for all f w in G one has (2) tail(f σ(w) ) = tail(f w ) · x a n for some integer a.…”

On pseudo symmetric monomial curves

“…Then, LM(f 2 ) = X α21 1 X 4 is divisible by X 1 . This leads to a contradiction as [4,Lemma 2.7] implies that the tangent cone is not Cohen-Macaulay. Similarly, when α 21 + α 3 > α 1 , LM(f 3 ) = X α1−α21−1 1 X 2 is divisible by X 1 .…”

“…Example 3.19. Let (α 21 , α 1 , α 2 , α 3 , α 4 ) = (4,7,3,4,9). Then (n 1 , n 2 , n 3 , n 4 ) = (97, 154, 87, 74).…”

Complete Intersection Monomial Curves and the Cohen—Macaulayness of Their Tangent Cones

“…. , t n d ]] is Cohen-Macaulay constitutes an important problem studied by many authors, see for instance [1], [6], [14]. The importance of this problem stems partially from the fact that if the associated graded ring is Cohen-Macaulay, then the Hilbert function of K[[t n1 , .…”